Low-defect-density gamma phase aluminum oxide substrates for heteroepitaxial synthesis

ABSTRACT

Aluminum oxide (Al 2 O 3 ) thin films having a high γ-phase purity and low defect density and methods for making the aluminum oxide thin films are provided. Also provided are epitaxial heterostructures that incorporate the aluminum oxide thin films as growth substrates and methods of forming the heterostructures. The Al 2 O 3  films are pure, or nearly pure, γ-Al 2 O 3 . As such, the films contain no, or only a very low concentration of, other Al 2 O 3  polymorph phases. In particular, the Al 2 O 3  films contain no, or only a very low concentration of, the θ-Al 2 O 3  polymorph phase.

REFERENCE TO GOVERNMENT RIGHTS

This invention was made with government support under 1720415 awarded bythe National Science Foundation. The government has certain rights inthe invention.

BACKGROUND

Polymorphism is a widespread phenomenon in metal oxides due to thesimilarity of valence, bonding configuration, and density among phaseswith different structures but identical compositions. The phenomenonoccurs in a wide range of materials and can arise in part from thecomplexity that accompanies the creation of compounds involving severalmetal ion species. Even comparatively chemically simple oxidesconsisting of a single metal ion, however, can have several metastablecrystal structures with corresponding differences in density, symmetry,and ionic coordination. These polymorphs can exhibit significantdifferences in properties relevant to their direct use in optical,electronic, magnetic, and catalytic applications and their indirectapplication as substrates for epitaxial growth of other compounds. Theexistence of multiple polymorphs leads to a range of kinetic phenomenain crystallization, especially in nanoscale systems in which thestructure is influenced by the proximity of surfaces and interfaces, andby complex elastic stress distributions.

The synthesis of phase-pure metastable oxides, especially in the form ofthin films, has been challenging because in the best cases thepolymorphs exhibit only small differences in free energy with respect tothe stable phase or other competing metastable phases. Conditions thatfavor the thermodynamically stable phase, such as high temperatures andlong annealing times, can make the metastable polymorphs highlyunfavorable and thus can make their formation even more challenging.Synthesis methods designed to yield metastable polymorphs of a selectedcompound thus often yield materials in which multiple polymorphs arepresent. The practical implications of this synthesis challenge are thatit is particularly difficult to produce metastable polymorphs with thedegree of control required to employ their promising properties inapplications and to conduct and to interpret experiments to determinetheir structure and properties.

Aluminum oxide, Al₂O₃, exhibits multiple polymorphs and has widespreadapplications in electronic materials, catalysis, and surfacepassivation. The polymorphs of Al₂O₃ include the thermodynamicallystable phase at ambient pressure, α-Al₂O₃, as well as several structuralphases that are metastable at ambient pressure.

The initial crystallization of amorphous Al₂O₃ often occurs through theformation of γ-Al₂O₃. It has been shown that planar thin films ofamorphous Al₂O₃ on Al₂O₃ crystallize into γ-Al₂O₃ upon heating andsubsequently convert to the α-Al₂O₃ at high temperature. solid-phaseepitaxy (SPE) of Al₂O₃ on sapphire substrates has previously beenstudied using amorphous Al₂O₃ created by methods including electron-beamevaporation, ion implantation, and atomic layer deposition (ALD). SPEleads to the formation of γ-Al₂O₃ from the amorphous layers, whichoccurs with a rate matching the duration of practical experiments athigh temperatures; e.g., 800° C. (Simpson, T. W. et al., J. Am. Ceram.Soc. 1998, 81, 61-66; Jang, J. et al., ACS Appl. Mater. Interfaces 2018,10, 41487-41496; McCallum, J. C. et al., Nucl. Instr. and Meth. in Phys.B 1994, 91, 60-62.) The γ-Al₂O₃ then transforms to the stable α-phase athigher temperature, often above 800° C. (Simpson et al., 1998; Jang etal., 2018; McCallum et al., 1994.)

Crystallized Al₂O₃ thin films produced from amorphous layers typicallyhave x-ray rocking curve angular widths that are far larger than singlecrystals, indicating that there is a comparatively high concentration ofdefects. (Levin, I. et al., J. Am. Ceram. Soc. 1998, 81, 1995-2012; andYu, N. et al., Phys. Rev. B 1995, 52, 17518-17522.) The high defectdensity reduces the usefulness of materials exhibiting multiple phases,for example in applications as substrates for subsequent epitaxialgrowth.

SUMMARY

Al₂O₃ thin films having a high γ-phase purity and low defect density andmethods for making the aluminum oxide thin films are provided. Alsoprovided are heterostructures that incorporate the Al₂O₃ thin films.

One embodiment of an Al₂O₃ film comprises at least 70 mol. % γ-Al₂O₃,the Al₂O₃ film having an x-ray rocking curve for an Al₂O₃ 222 reflectionwith a full width at half maximum of 0.06° or lower, a film thickness ofless than 200 nm, and a lateral cross-sectional area of at least 1 cm².

Once embodiment of a heterostructure includes: an Al₂O₃ film comprisingat least 70 mol. % γ-Al₂O₃, the Al₂O₃ film having an x-ray rocking curvefor an Al₂O₃ 222 reflection with a full width at half maximum of 0.06°or lower, a film thickness of less than 200 nm, and a lateralcross-sectional area of at least 1 cm²; and an epitaxial overlayer onthe Al₂O₃ film, the epitaxial overlayer comprising an inorganic thinfilm.

Other principal features and advantages of the invention will becomeapparent to those skilled in the art upon review of the followingdrawings, the detailed description, and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Illustrative embodiments of the invention will hereafter be describedwith reference to the accompanying drawings, wherein like numeralsdenote like elements.

FIG. 1A shows amorphous Al₂O₃ on α-Al₂O₃. FIG. 1B shows epitaxialγ-Al₂O₃ on α-Al₂O₃. FIG. 1C shows epitaxial θ-Al₂O₃ on α-Al₂O₃. The unitcells of each polymorph appear as solid lines. FIG. 1D showsreciprocal-space locations of x-ray reflections from γ-Al₂O₃ and θ-Al₂O₃used in the experiments described in the Example.

FIG. 2A shows the scattering geometry for x-ray experiments probingγ-Al₂O₃ 400 and θ-Al₂O₃ 600 reflections, as described in the Example.The vectors X-ray k_(in) and X-ray k_(out) denote the directions andmagnitudes of the wavevectors of the incident x-ray beam and the beamdiffracted to the center of the detector, respectively. FIG. 2B showsintegrated intensities of θ-Al₂O₃ 600 and γ-Al₂O₃ 400 reflectionsobtained after heating to different temperatures. FIG. 2C showsfractions of γ-Al₂O₃ and θ-Al₂O₃ as a function of heating temperature.

FIG. 3A shows marginal and joint distribution of q _(6,Al—Al) and q_(6,O—O) from molecular dynamics (MD) simulations of amorphous Al₂O₃,termed A in the legend, and the α-Al₂O₃, γ-Al₂O₃, and θ-Al₂O₃ polymorphsat 1600 K and 1 atm. Atomic positions and the position dependence of q_(6,Al—Al) and q _(6,O—O) for (FIG. 3B) the amorphous Al₂O₃ on α-Al₂O₃starting configuration and (FIG. 3C) the γ-Al₂O₃ on α-Al₂O₃configuration reached after crystallization.

FIG. 4A shows the scattering geometry for x-ray experiments probing the202 and 401 reflections of θ-Al₂O₃, as discussed in the Example.Diffraction patterns exhibiting 202 and 401 reflections of θ-Al₂O₃ afterheating to (FIG. 4B) 800° C. and (FIG. 4C) 1020° C. FIG. 4D showsintensities as a function of χ obtained by integrating across theindicated regions of FIG. 4B and FIG. 4C. Notations A and B indicatewhich of two variants of θ-Al₂O₃ contributes to each reflection, suchthat 401A and 202A arise from variant A. The sharp features near χ=0arise from scattering along the surface-normal reciprocal spacetruncation rod.

FIG. 5A shows a θ/2θ x-ray diffraction scan along the surface normaldirection of reciprocal space after heating to 800° C. or 1020° C. for 1hour. FIG. 5B shows rocking curves of the γ-Al₂O₃ 222 reflection forfilms heated to 800° C. and 1020° C. for 1 h.

FIG. 6A shows scattering geometry for x-ray experiments probing γ-Al₂O₃400 and θ-Al₂O₃ 600 reflections. FIG. 6B shows an X-ray diffractionpattern recorded using the geometry indicated in FIG. 6A during arocking scan after heating to 800° C. FIG. 6C shows intensity as afunction of 2θ obtained by integrating 2.5°-wide range of χ angles nearthe γ-Al₂O₃ 400 reflection. FIG. 6D shows an X-ray diffraction patternrecorded during a rocking scan after heating to 1020° C. FIG. 6E showsintensity as a function of 2θ obtained by integrating the same range asFIG. 6C, exhibiting the γ-Al₂O₃ 400 and θ-Al₂O₃ 600 reflections.

FIGS. 7A-7B show bonding configurations and epitaxial relationships atthe interfaces of (FIG. 7A) epitaxial γ-Al₂O₃ on α-Al₂O₃, and (FIG. 7B)epitaxial θ-Al₂O₃ on α-Al₂O₃.

FIGS. 8A-8C show representative structures of γ-Al₂O₃ in which (FIG. 8A)all Al vacancies occupy octahedral sites, (FIG. 8B) all Al vacanciesoccupy tetrahedral sites, and (FIG. 8C) the occupation has the samestatistics as the site occupation matching the experimentally observedlattice parameter. The inset shows the local configuration of octahedraland tetrahedral Al sites.

DETAILED DESCRIPTION

Al₂O₃ thin films having a high γ-phase purity and low defect density andmethods for making the aluminum oxide thin films are provided. Alsoprovided are epitaxial heterostructures that incorporate the aluminumoxide films as growth substrates and methods of forming the epitaxialheterostructures.

The Al₂O₃ films are pure, or nearly pure, γ-Al₂O₃ and have a low densityof defects. As such, the films contain no, or only a very lowconcentration of, other Al₂O₃ polymorph phases. In particular, the Al₂O₃films contain no, or only a very low concentration of, the γ-Al₂O₃polymorph phase, which is naturally present as a competing phase inAl₂O₃ films that are crystallized using previously reported methods.Various embodiments of the highly γ-Al₂O₃ phase pure, low defect densityfilms comprise at least 70 mol. % γ-Al₂O₃. This includes embodiments ofthe highly γ-Al₂O₃ phase pure, low defect density films comprising atleast 90 mol. % γ-Al₂O₃, and at least 98 mol. % γ-Al₂O₃. For example,the films can have γ-Al₂O₃ phase fractions in the range from 90 mol. %to 100 mol. % γ-Al₂O₃ and from 95 mol. % to 98 mol. % γ-Al₂O₃.

The ability to form the high γ-phase purity and low defect density Al₂O₃films is based, at least in part, on the inventors' discovery thatamorphous Al₂O₃ films on a α-Al₂O₃ sapphire substrate initiallytransform, upon heating, into epitaxial γ-Al₂O₃, followed by atransformation to an intermediate monoclinic γ-Al₂O₃ phase, andeventually to α-Al₂O₃. The discovery that an intermediate γ-Al₂O₃ phaseforms during the transformation from γ-Al₂O₃ to α-Al₂O₃ and a detailedstudy of the kinetics of the epitaxy and transitions between themetastable phases of Al₂O₃ during the transformation enabled theidentification of SPE conditions that result in the crystallization ofthe γ-Al₂O₃ phase with a drastically reduced density of structuraldefects from amorphous Al₂O₃, while avoiding the microstructuralinhomogeneity arising from the spatially inhomogeneous transformation toγ-Al₂O₃.

Past studies have overlooked the presence of the γ-Al₂O₃ phase duringthe multi-step transformation, likely because it is very difficult todistinguish the x-ray reflections from the different phases usingmeasurements of the diffracted intensity along the surface normal, i.e.,conventional thin-film x-ray θ/2θ scans. Similarly, there are severalother strong reflections at which the γ-Al₂O₃ and γ-Al₂O₃ areindistinguishable. As a result, the γ-Al₂O₃ phase has not been reportedin studies of the crystallization of amorphous Al₂O₃ on α-Al₂O₃ and, asa result, no steps have been taken in the reported studies to avoid theformation of the γ-Al₂O₃ phase.

The γ-phase purity and low defect density of the Al₂O₃ films describedherein are reflected in the narrow width, as measured by the full widthat half maximum (FWHM) of the x-ray rocking curve for the Al₂O₃ 222reflection. By way of illustration, various embodiments of the Al₂O₃films have x-ray rocking curves for the Al₂O₃ 222 reflection with FWHMsof 0.06° or lower. These include Al₂O₃ films having x-ray rocking curvesfor the Al₂O₃ 222 reflection with FWHMs of 0.05° or lower, 0.04° orlower, and 0.03° or lower. For example, Al₂O₃ films having x-ray rockingcurves for the Al₂O₃ 222 reflection with FWHMs in the range from 0.01°to 0.06°, in the range from 0.02° to 0.05°, and in the range from 0.03°to 0.05° can be formed using the methods described herein. (Methods formeasuring the FWHM of an x-ray rocking curve for a Al₂O₃ 222 reflectionare described in the Example.) In contrast, the FWHM of the rockingcurves for γ-Al₂O₃ films reported in the literature are ten to twentytimes larger—indicating the presence of substantial concentrations ofadditional polymorphs, such as θ-Al₂O₃.

Films of highly γ-Al₂O₃ phase pure aluminum oxide with low defectdensities, as reflected by low mosaic widths in their x-ray rockingcurves, can be formed by a kinetic route using SPE. SPE is thecrystallization from an amorphous form with an orientation or phaseselected using a seed crystal. The crystallization and phasetransformation in SPE occur through the atomic reconfiguration atinterfaces.

SPE can be carried out on an amorphous Al₂O₃ film, which initiallytransforms upon heating to form epitaxial γ-Al₂O₃. The transformation isthen arrested to prevent or minimize the growth of the metastablemonoclinic θ-Al₂O₃ phase and, eventually, the α-Al₂O₃ phase. Thestarting amorphous Al₂O₃ film may itself be deposited on a (0001)α-Al₂O₃ sapphire substrate. However, α-Al₂O₃ substrates with othersurface orientations, such as (1210) and (0112), can also be used. Theorientations of the γ-Al₂O₃ and epitaxial films grown on it can beextended to (001) due to the surface symmetries of other α-Al₂O₃substrates.

Illustrative steps in an SPE protocol that can be used to form the lowdefect density, highly γ-Al₂O₃ phase pure films desirably begins withpre-treatment to remove contaminants from the α-Al₂O₃ substrate. Suchpretreatments generally include a washing step and a thermal treatment.By way of illustration only, the α-Al₂O₃ substrate may be sonicated indeionized (DI) water, then heated to a temperature above 1200° C. (e.g.,˜1400 C) for a period of 6 hours or more. A layer of amorphous Al₂O₃ isthen ° formed on the α-Al₂O₃ substrate. The amorphous layer can beformed using, for example, atomic layer deposition (ALD). Once formed,the ° amorphous Al₂O₃ layer is heated to a temperature and for a timesufficient to induce the transformation of the amorphous Al₂O₃ into afilm of γ-Al₂O₃ that is free of, or contains only very lowconcentrations of other Al₂O₃ polymorphs, such as θ-Al₂O₃. Suitablecrystallization temperatures and times are discussed below andillustrated in the Example. The SPE crystallization can be carried outunder an atmosphere that does not affect the crystallization process,such as in O₂ gas or air and may be, but is not necessarily, conductedat atmospheric pressure.

The γ-Al₂O₃ films grown via SPE on α-Al₂O₃ sapphire can include a singledomain, or two domains related by a 180° rotation around [111]. Thepresence of two domains is consistent with the two possible ways thatthe face-centered cubic (FCC) stacking sequence of the oxygen layers canbe continued from the α-Al₂O₃ substrate into the γ-Al₂O₃ layer. However,a single-domain γ-Al₂O₃ film can be grown by using a substrate in whicha surface miscut orientation and magnitude, on the order of degrees,promote the formation of a single domain.

An important factor in preventing the formation of the θ-Al₂O₃ phaseduring SPE is the crystallization temperature, with lower temperaturesfavoring a highly phase pure γ-Al₂O₃ film. Generally, temperatures of700° C. or lower promote the transformation to γ-Al₂O₃ without thegrowth of the θ-Al₂O₃ phase. Temperatures of less than 700° C. may beused. However, the rate of crystallization will decrease with decreasingtemperatures. By way of illustration only, crystallization temperaturesin the range from 680° C. to 700° C. may be used. However, highertemperatures can be used for thinner films.

Other conditions that should be carefully controlled to avoid the growthof the θ-Al₂O₃ polymorph are the SPE crystallization time and the filmthickness. If the crystallization is allowed to proceed too long, theθ-Al₂O₃ phase will begin to form. Therefore, SPE crystallization timesof 60 minutes or shorter are suitable. For example, SPE crystallizationtimes in the range from 10 minutes to 30 minutes can be used. Low filmthicknesses are advantageous because thin films allow the amorphousAl₂O₃ film to crystallize into γ-Al₂O₃ throughout the entire thicknessof the film before the onset of θ-Al₂O₃ phase growth. In addition,thinner films may have a different stress state and may include fewerstructural defects arising from processes that relax the epitaxialmismatch. The reduced defect concentration can result in fewer defectson which the θ-Al₂O₃ phase can nucleate. For this reason, the low defectdensity, highly γ-Al₂O₃ phase pure films and the amorphous Al₂O₃ filmsfrom which they are formed desirably have film thicknesses of 200 nm orless, including thicknesses of 100 nm or less, 50 nm or less, or 10 nmor less. By way of illustration, various embodiments of the low defectdensity, highly γ-Al₂O₃ phase pure films have thicknesses in the rangefrom 0.8 nm to 200 nm, including the range from 0.8 nm to 100 nm, andfurther including the range from 0.8 nm to 50 nm and from 0.8 nm to 10nm. γ-Al₂O₃ films with thicknesses in these ranges having a γ-Al₂O₃phase purity of at least 70 mol. %, at least 90 mol. %, and at least 98mol. % can be formed.

These illustrative temperature, time, and film thickness parameters areinterrelated and, therefore, can be adjusted together to produce a lowdefect density, highly γ-Al₂O₃ phase pure film. For example, at lowercrystallization temperatures, longer crystallization times and/orthicker films can be produced. Alternatively, low defect density, highlyγ-Al₂O₃ phase pure films with very small film thicknesses can beproduced using higher crystallization temperatures and/or longercrystallization times. Thus, for very thin films, temperatures greaterthan about 700° C. can be used, including temperatures in the range from700° C. to 800° C. At the higher end of this temperature range (forexample, from 750° C. to 800° C.) highly γ-Al₂O₃ phase pure, low defectdensity films having thicknesses of less that 20 nm, includingthicknesses of less than 10 nm can be formed. Such films include thosehaving a γ-Al₂O₃ phase content of greater than 95 mol. %, greater than99 mol. % and 100 mol. %.

Large-area films of the highly phase-pure γ-Al₂O₃ can be formed. Thismakes the films suitable for use as growth substrates for the epitaxialgrowth of large-area inorganic oxide thin films of compounds havingcubic or hexagonal crystal structures. By way of illustration, highlyphase-pure γ-Al₂O₃ films having a lateral cross-sectional area of atleast 1 cm 2, at least 2 cm², at least 10 cm², or at least 100 cm² canbe produced, where the lateral cross-section refers to a planeorthogonal to the film thickness direction. Perovskite oxides and/orother materials that have been grown on SrTiO₃ substrates are examplesof materials that can be grown epitaxially on the low defect density,highly phase-pure γ-Al₂O₃ films. The replacement of SrTiO₃ with γ-Al₂O₃as a growth substrate is advantageous because γ-Al₂O₃ has much lowerdielectric loss at high frequencies compared with SrTiO₃. The epitaxialgrowth of one or more overlayers may be carried out using known methods,such as chemical vapor deposition or molecular beam epitaxy.

Some potential application areas of the highly γ-Al₂O₃ phase pure, lowdefect density films, including the (111)-oriented films, are asepitaxial growth substrates for Pb[Zr_(x)Ti_(1-x)]O₃ (lead zirconate;commonly referenced as PZT), which has valuable applications inultrasound and integrated optics applications; and BiFeO₃ (bismuthferrite; commonly referenced as BFO), which has been developed as apotential lead-free piezoelectric material. Notably, α-Al₂O₃ substratesare well-suited for the growth of BFO films because the thermalexpansion coefficient of sapphire is close to that of BFO. However, itis difficult to realize the epitaxial growth of BFO films directly onα-Al₂O₃ because of the different structure symmetry and the largelattice mismatch. The highly phase pure γ-Al₂O₃ films described hereincan promote the crystallization and orientation control of the BFO thinfilms due to the similarity in crystal structures. Other inorganiccompounds that can be grown on the highly γ-Al₂O₃ phase pure, low defectdensity films include KTaO₃ (potassium tantalate; commonly referenced asKTO). KTO is a quantum paraelectric material, which is on the verge ofthe ferroelectric transition in which ferroelectricity is compressed byquantum fluctuations at low temperatures. Superconductivity has beendiscovered at the interface between (111)-oriented KTO and LaAlO₃ orEuO. Thus, the high γ-Al₂O₃ phase purity, low defect density films withan (111) orientation described herein can be used to synthesize(111)-oriented KTO-based heterostructures with superconductinginterfaces. SrVO₃ (strontium vanadate; commonly referenced as SVO) canalso be grown epitaxially on the highly γ-Al₂O₃ phase pure, low defectdensity films. SVO is a transparent conducting oxide that can be used toform electrodes for photovoltaic cells and displays and to create smartwindows. Transparent conducting SVO thin films with a cubic crystalstructure can be synthesized on the highly γ-Al₂O₃ phase pure, lowdefect density films. In addition, SVO is water-soluble and SVO bufferlayers can be used to synthesize large-area freestanding thin films onthe highly γ-Al₂O₃ phase pure, low defect density films by depositingSVO thin films thereon before the deposition of other thin films. TheSVO buffer layer then can be dissolved in water and the released toplayer will be a free-standing thin film with a large area. Garnets areanother example of materials that can be grown epitaxially on the highlyγ-Al₂O₃ phase pure, low defect density films. Garnets have the generalchemical formula X₃Z₂(TO₄)₃, where X, Y, and T are metals or metalloids.The lattice parameters of garnets are commonly approximately 12 Å andcan be epitaxially grown on a 7.88 Å lattice parameter of γ-Al₂O₃through an arrangement in which two garnet unit cells span the samedistance as three γ-Al₂O₃ unit cells. Garnets have applications inmagnetooptical devices and as emerging spintronic and spin caloritronicdevices requiring large area thin films.

Example

The experiments and simulations reported here include detailedstructural characterization and MD simulations of the initial stages ofthe crystallization of amorphous Al₂O₃ and the kinetic regimes in whichγ-Al₂O₃ and γ-Al₂O₃ are favored. The results reveal that γ-Al₂O₃ can beformed with minimal formation of the competing polymorphs. The initiallyamorphous Al₂O₃ film on (0001) α-Al₂O₃ substrate is shown schematicallyin FIG. 1A. The structures of the crystallized γ-Al₂O₃ and the θ-Al₂O₃layers formed by the transformation from γ-Al₂O₃ are shown in FIGS. 1Band 1C, respectively. The epitaxial relationships depicted in FIGS.1A-1D are consistent with the experimental results, and the structuralprinciples guiding the proposed interface configuration are describedbelow.

The interpretation of the x-ray diffraction results was guided byconsidering the reciprocal-space locations of the x-ray reflections ofγ-Al₂O₃ and θ-Al₂O₃. FIG. 1D illustrates the reciprocal-space locationsof several γ-Al₂O₃ and θ-Al₂O₃ x-ray reflections considered in thisexample. The diagram in FIG. 1D includes reflections for two variants ofγ-Al₂O₃ and six structural variants of θ-Al₂O₃, with reciprocal-spacelocations in the experimental observations below. The γ-Al₂O₃ andθ-Al₂O₃ x-ray reflections along the surface-normal direction ofreciprocal space were not widely separated. The 201 reflection ofθ-Al₂O₃ and the 111 reflection of γ-Al₂O₃, for example, had interplanarspacings that differed by only 0.05 Å. The small differences made itdifficult to distinguish these reflections using measurements of thediffracted intensity along the surface normal, i.e., conventionalthin-film x-ray θ/2θ scans. Similarly, there were several other strongreflections at which the γ-Al₂O₃ and θ-Al₂O₃ were indistinguishable.These reflections have been studied in previous reports, and as aresult, the θ-Al₂O₃ phase has not been reported in studies of thecrystallization of amorphous Al₂O₃ on α-Al₂O₃. (Simpson et al., 1998;Jang et al., Clarke, D. R., Phys Status Solidi A 1998, 166,183-196.2018; Yu, N. et al., Appl. Phys. Lett. 1995, 67, 924-926;McCallum et al., 1994; Sklad, P. S. et al., Nucl Instrum Meth B 1990,46, 102-106; White, C. W. et al., Nucl. Instr. and Meth. in Phys. Res. B1988, 32, 11-22.) The experiments described herein employed two sets ofreflections in which γ-Al₂O₃ and θ-Al₂O₃ can be separately probed, asillustrated in FIG. 1D. Reflections in the γ-Al₂O₃ 400 family andθ-Al₂O₃ 600 family were close together but can be distinguished withinthe resolution of thin-film x-ray diffraction experiments and theangular width of the reflections of practical samples.

Crystallized Al₂O₃ thin films produced from amorphous layers typicallyhave x-ray rocking curve angular widths that are far larger than singlecrystals, indicating that there is a comparatively high concentration ofdefects. (Levin, I. et al., J. Am. Ceram. Soc. 1998, 81, 1995-2012;Simpson et al., 1998; Yu, N. et al., Phys. Rev. B 1995, 52, 17518-17522;McCallum et al., 1994; White et al., 1988.) The high defect densityreduces the usefulness of materials exhibiting multiple phases, forexample in applications as substrates for subsequent epitaxial growth.In addition, high-resolution thin-film x-ray diffraction measurements ofthe lattice parameters and microstructural features are impractical inpoorly ordered structures. The results here indicate that the phasetransformation between polymorphs and the resulting structuralinhomogeneity contributed significantly to this mosaic broadening. Thefull width at half maximum (FWHM) of the rocking curves of γ-Al₂O₃ 222x-ray reflection under these conditions was 0.03°, indicatinglow-mosaicity in the film. FWHM value was much smaller than the bestvalue reported in previous studies without such control, 1.4°. (Yu etal., 1995.) The discovery of this phase sequence in epitaxial thin filmsallowed low-defect-density γ-Al₂O₃ to be formed under conditions with alow concentration of γ-Al₂O₃. The large single-crystalline γ-Al₂O₃ canbe used as substrates for materials with cubic or hexagonal structures,which can broaden the choices of substrates for oxides and enable largearea processing on low-defect-density commercial substrates.

Methods

Deposition and Heating

Amorphous Al₂O₃ films with thicknesses of 100 nm were deposited on(0001) α-Al₂O₃ substrates by atomic layer deposition (ALD). Thesubstrates were prepared by sonication in DI water for 40 min andheating to 1400° C. in air for 10 h. These processes yielded a steppedsurface morphology with uniform terrace width, as observed using atomicforce microscopy. The substrate was cooled to 1000° C., removed from thefurnace, and immediately loaded into the ALD reactor under flowinghigh-purity N₂ gas. Amorphous thin films deposited on substrates whenthere was a significant delay between the introduction to the ALDreactor resulted in poorly ordered crystalline after heating, suggestingthat the interfacial contamination between the substrates and the thinfilms should be controlled in order to obtain high quality thin films.The ALD reactor was held at 200° C. during the deposition. The ALDprocess used a series of gas pulses: (i) trimethyl aluminum (in N₂carrier gas) for 5 s at 0.7 Torr, (ii) N₂ purge for 20 s at 0.5 Torr,(iii) H₂O (in N₂ carrier gas) for 18 s at 1.5 Torr, and (iv) N₂ purgefor 30 s at 0.5 Torr. The deposition rate for Al₂O₃ was 1.1 Å/cycle. Theamorphous films reported here were obtained with layers deposited usinga total of 850 cycles.

The amorphous Al₂O₃ thin films were crystallized and transformed betweenpolymorphs by heating in 02 gas at atmospheric pressure for 1 h attemperatures from 700° C. to 1020° C. After heating, the samples wereremoved from the furnace and cooled in room-temperature air atatmospheric pressure.

The scaling of the SPE crystallization process to lower Al₂O₃ layerthicknesses was then demonstrated through the crystallization of anamorphous layer with an initial thickness of 9.7 nm. A (0001) orientedα-Al₂O₃ substrate was prepared by chemical cleaning and heating in anoxidizing environment. An amorphous Al₂O₃ layer was deposited using theatomic layer deposition from trimethyl aluminum and water precursors.The thickness of the amorphous layer was measured using x-rayreflectivity, giving a thickness value of 9.7 nm. The Al₂O₃ layer wasfully crystallized by heating to 800° C. for 1 hour. The thicknessmeasured after the heating step using x-ray reflectivity was 8.1 nm. Thechange in the thickness indicated that the Al₂O₃ layer was crystallizedduring this heating process. Atomic force microscopy images of thesurface of the Al₂O₃ layer after heating indicated that it remainedsufficiently smooth for epitaxial growth after crystallization.

Characterization

A high-resolution diffractometer (Empyrean, Panalytical Inc.) was usedfor both x-ray reflectivity and high-resolution x-ray diffraction. Theincident beam optics employed either a multilayer mirror that yieldshigh-intensity Cu Kα radiation or a combination of the multilayer mirrorand a channel-cut crystal that yields monochromatic Cu Kai radiation. Asecond diffractometer (D8 ADVANCE, Bruker, Inc.) employed a Cu Kαpoint-focus beam and an area x-ray detector was used to determine thepolymorph composition and orientation. The measurements were taken atroom temperatures (i.e., ˜ 23° C.).

Molecular Dynamics (MD) Simulation Methods

Classical MD simulation probed the epitaxial crystallization ofamorphous Al₂O₃. The interatomic potential employed in these simulationshad previously been calibrated using the experimental lattice parametersof α-Al₂O₃ and the nearest neighbor Al-O distance. (Matsui, M.,Mineralog. Mag. 1994, 58, 571-572.) The amorphous structure producedusing this potential also agreed with x-ray and neutron diffractionobservations. (Gutiérrez, G. et al., Phys. Rev. B 2002, 65, 104202.) Theinteratomic potentials take the Coulomb-Buckingham form in which theshort-range electrostatics, the exponential repulsion, and van del Waalsattraction terms are cut off at 8.0 Å. The particle-mesh Ewald methodwas used to compute the long-range electrostatic interaction.

The MD simulation was conducted using the GROMACS 2018 program.(Berendsen, H. J. et al., Comp. Phys. Commun. 1995, 91, 43-56.) Thetemperature was set using a stochastic-term-based velocity-rescalingthermostat with a 0.1 ps relaxation time. Amorphous Al₂O₃ at selectedtemperatures and densities was generated using the following procedure.An α-Al₂O₃ crystal containing 4320 atoms (864 formula units) was firstequilibrated at 1600 K at 1 atm using the Berendsen barostat in theisothermal-isobaric ensemble. All simulations after the initialequilibration of α-Al₂O₃ were conducted in the canonical ensemble. Afterthe equilibration, the system was heated to 5000 K and then quenched to3000 K at a rate of 20 K/ps, which removed the dependence of the finalamorphous structure on the starting polymorph. The density of theamorphous Al₂O₃ system was changed to 3.0 g/cm³, approximately matchingthe experimentally observed density, by rescaling atomic positions andthe size of the simulation box along the direction that was subsequentlyused as the surface normal during simulations of epitaxialcrystallization. The simulation box dimensions along the other twodirections were not adjusted in order to allow the amorphous structureto be joined to the substrate in a subsequent step. The final amorphousAl₂O₃ was obtained by quenching the system from 3000 K to 1600 K at 2K/ps and equilibrating for 200 ps after quenching.

The amorphous-crystalline interface was created by appending theamorphous structure to the (0001) surface of an equilibrated α-Al₂O₃supercell containing 2160 atoms serving as the substrate for epitaxialcrystallization. A vacuum layer with a thickness of 30 Å was added ontop of the amorphous structure. A two-step relaxation was employedbefore the production run to remove artifacts in the simulatedamorphous/α-Al₂O₃ interface structure. An initial relaxation step wasperformed in which the α-Al₂O₃ substrate was subjected to a strongposition constraint so that only the amorphous layer relaxed. In asecond step, all atoms except the ones within 8 Å of theamorphous-crystalline interface were subject to the same positionconstraint so that atoms near the interface were fully relaxed. Bothrelaxations employed durations of 200 ps at 1600 K.

The production run had a simulated duration of 20 ns. A layer of α-Al₂O₃with a thickness of 5 Å at the bottom of the simulation box, farthestfrom the interface, was frozen in order to mimic a semi-infinitesubstrate. The results below are analyzed using a coordinate system inwhich the z-axis is along the [0001] direction of the crystallineα-Al₂O₃ substrate.

The results of the MD simulation were studied using a statisticalanalysis of the coordination of the Al and 0 atoms. The Steinhardt orderparameter q _(lβ-β)(i) was used to distinguish between differentpolymorphs of Al₂O₃. (Lechner, W. et al., J. Chem. Phys. 2008, 129,114707.) The quantity q _(l,β-β)(i) was computed by first defining aquantity q_(lm,β-β)(i) to measure the coordination symmetry up to thefirst coordination shell:

$\begin{matrix}{{q_{{lm},{\beta - \beta}}(i)} = {\frac{1}{N_{\beta}(i)}{\sum_{j = 1}^{N_{\beta}(i)}{Y_{lm}\left( r_{ij} \right)}}}} & (1)\end{matrix}$

Here the index i enumerates all atoms of type β, N_(β)(i) is the numberof nearest neighbors of type β around the atom with index i, and 1 and mare integer parameters where m runs from −l to l. The functionsY_(lm)(r_(ij)) are spherical harmonics and r_(ij) is the unit vectorfrom atom i to atom j. A second quantity, q _(lm,β-β)(i), providesinformation on the coordination symmetry up to the second coordinationshell by averaging of q_(lm,β-β)(i) over its neighbors of type β:

$\begin{matrix}{{{\overset{\_}{q}}_{{lm},{\beta - \beta}}(i)} = {\frac{1}{1 + {N_{\beta}(i)}}\left\lbrack {{q_{{lm},{\beta - \beta}}(i)} + {\sum_{j = 1}^{N_{\beta}(i)}{q_{{lm},{\beta - \beta}}(j)}}} \right\rbrack}} & (2)\end{matrix}$

Here the sum includes atom i and its neighbors of type β. Taking theaverage of q_(lm,β-β)(i) provides a clearer distinction betweendifferent crystal structures at the price of losing coordinationinformation around single atoms. The Steinhardt bond order parameter q_(l,β-β)(i) for atom i is obtained by summing over m, yielding arotationally invariant result ranging from 0 to 1 that is useful inrecognizing the crystal structure regardless of the spatial orientation:

$\begin{matrix}{{{\overset{\_}{q}}_{l,{\beta - \beta}}(i)} = \sqrt{\frac{4\pi}{{2l} + 1}{\sum_{m = {- l}}^{l}{❘{{\overset{\_}{q}}_{{lm},{\beta - \beta}}(i)}❘}^{2}}}} & (3)\end{matrix}$

The Steinhardt order parameter is sensitive to different coordinationsymmetries depending on the choice of 1. (Steinhardt, P. J. et al.,Phys. Rev. B 1983, 28, 784-805.) The oxygen sublattice in the polymorphsof Al₂O₃ discussed in this example is either hexagonal close packed, asfor α-Al₂O₃, or FCC, as in γ-Al₂O₃ and γ-Al₂O₃. Both symmetries can beprobed by the l=6 spherical harmonics. Hence q _(6,O—O) and q _(6,Al—Al)were used in this example to identify different polymorphs of Al₂O₃.

The coordination number distribution of 0 atoms around Al atoms can bedetermined by computing the fraction of various Al-0 polyhedra in whichAl is coordinated with χ=3, 4, 5, or 6 O ions. The x=4 and x=6 casescorresponded to tetrahedral and octahedral coordination, respectively.An 0 atom was considered as coordinated to the central Al atom if theirseparation is within a hard cutoff r_(c)=2.5 Å, which is the firstminimum of the partial radial distribution function between Al atoms andO atoms. In this example, coordination number distribution of O atomsaround Al atoms for simulated amorphous Al₂O₃ and epitaxial γ-Al₂O₃ werecomputed to determine the distribution of Al atoms over tetrahedral andoctahedral spinel sites.

Results and Discussion

Polymorph Identification and Phase Fractions of γ-Al₂O₃ and θ-Al₂O₃

The crystallization of amorphous Al₂O₃ resulted in the development of asequence of crystalline phases. The mass density, as evaluated usingx-ray reflectivity, provided a guideline for the overall progress of thecrystallization. The density of the as-deposited amorphous Al₂O₃ layersdetermined from the critical angle for total external x-ray reflectionwas 3.1±0.1 g/cm³. This amorphous layer density matched the valuereported in the literature. (Groner, M. et al., Chem. Mater. 2004, 16,639-645.) The crystallization of Al₂O₃ resulted in a decrease of thefilm thickness and an increase in density, both of which were apparentin x-ray reflectivity measurements. Al₂O₃ layers heated to acomparatively low temperature of 750° C. were not crystallized after 1 hand exhibited a decrease in thickness of only 3%, corresponding to asmall increase with respect to the as-deposited density. Heating tobetween 800° C. and 1020° C. for 1 h resulted in a decrease in thethickness by 16% to a density of 3.5±0.1 g/cm³, consistent with thecrystallization of the Al₂O₃ layer. The densities of γ-Al₂O₃ (3.61g/cm³) and γ-Al₂O₃ (3.60 g/cm³) could not be distinguished within theprecision of the x-ray reflectivity measurement.

The dependence of phase composition of the Al₂O₃ layer on heatingtemperature was studied using a series of x-ray diffractionmeasurements. The x-ray characterization employed a region of reciprocalspace in which reflections in the γ-Al₂O₃ 600 family and the γ-Al₂O₃ 400family appeared and can be clearly distinguished, as illustrated in FIG.2A. The intensities I_(γ400) and I_(θ600) for the γ-Al₂O₃ 400 andγ-Al₂O₃ 600 families, respectively, were both obtained by integratingand averaging the intensities of the six orientations of eachreflection. The intensities obtained in this way are shown in FIG. 2Bfor layers heated to a series of different temperatures.

A quantitative analysis of the intensities in FIG. 2B allows the volumefractions of the γ-Al₂O₃ and γ-Al₂O₃ phases to be determined. Withvolumes V_(γ) and V_(θ) for the two phases, the volume fractions ƒ_(γ)and ƒ_(θ) are

${f_{\gamma} = {\frac{V_{\gamma}}{V_{\theta} + V_{\gamma}} = \frac{I_{\gamma 400}}{I_{{tot},{corr}}}}}{and}{{f_{\theta} = {\frac{V_{\gamma}}{V_{\theta} + V_{\gamma}} = \frac{{cI}_{\theta 600}}{I_{{tot},{corr}}}}},}$

with I_(tot,corr)=I_(γ400)+cI_(θ600). Here

${c = {{\frac{{❘F❘}_{\gamma 400}^{2}}{{❘F❘}_{\theta 600}^{2}}\frac{{LP}_{\gamma 400}}{{LP}_{\theta 400}}\frac{N_{\theta 600}}{N_{\gamma 400}}} = 7.69}},$

where F, with appropriate subscripts, is the structure factor for eachset of reflections. LP is Lorentz-polarization factor for the twophases. Nis the number of variants that contribute to the integratedintensity: 2 for the γ-Al₂O₃ 400 reflections and 6 for γ-Al₂O₃ 600reflections. The volume fractions of γ-Al₂O₃ and γ-Al₂O₃ are shown as afunction of temperature in FIG. 2C.

The intensities of reflections for both phases were zero, withinexperimental uncertainty, following heating to 750° C. Two othertemperature regimes are apparent in FIGS. 2B and 2C. After heating to800 and 850° C., the majority of the crystallized layer was in theγ-Al₂O₃ phase, with ƒ_(θ)=30%. Higher temperatures during the heatingprocess, from 900° C. to 1020° C., resulted in higher values of ƒ_(θ),approximately 80% after heating to 1020° C. Scanning transmissionelectron microscopy measurements were conducted for films heated to1020° C. and the results indicated that both γ-Al₂O₃ and γ-Al₂O₃appeared in the first few tens of nm from the interface and wereapproximately randomly distributed in the layer crystallized under theseconditions.

A further experiment was conducted to determine whether γ-Al₂O₃ formeddirectly from amorphous Al₂O₃ or resulted from a transformation fromγ-Al₂O₃. An initially amorphous Al₂O₃ layer was heated at 800° C. for 1h to form γ-Al₂O₃ and then heated again to 1020° C. for a further 1 h.The fraction of γ-Al₂O₃ following this two-step process was 80%, whichis the same as the fraction observed in the film directly heated to1020° C. for 1 hour. It was thus concluded that the γ-Al₂O₃ resultedfrom a transformation from the γ-Al₂O₃ rather than a directcrystallization from the amorphous layer into θ-Al₂O₃.

The Steinhardt order parameter q _(6,Al—Al) and q _(6,O—O) was used toidentify the polymorphs of Al₂O₃ formed during crystallization in MDsimulations. Reference values of these parameters were obtained fromsimulations of the bulk phases at 1600 K and 1 atm. The probabilitydistribution of q _(6,Al—Al) and {circumflex over (q)}_(6,O—O) and theirjoint distribution for four different Al₂O₃ structures reached afterdifferent stages in the MD simulation are shown in FIG. 3A. The marginaldistributions were normalized on the unit interval [0,1] and the jointdistributions were normalized on the unit region [0,1]×[0,1]. γ-Al₂O₃and O—Al₂O₃ had nearly identical q _(6,O—O) distributions with a peakvalue of 0.32, because the O sublattices in both γ-Al₂O₃ and γ-Al₂O₃exhibited FCC packing. The most common Al-O coordinations were those inwhich Al is coordinated with 4 O ions (47%) and 5 O ions (47%). A smallproportion, approximately 10%, of the ions were in three-fold andsix-fold coordinations. In the epitaxial γ-Al₂O₃ layers produced by theMD simulation, the most frequent Al coordinations were tetrahedralcoordination (29%) and octahedral coordination (70%). Amorphous Al₂O₃and γ-Al₂O₃ also had similar distributions of values of q _(6,Al—Al)with maxima at 0.12 and 0.15, respectively. The amorphous and γ-Al₂O₃structures had Al ion arrangements that were more similar to each otheras compared to other polymorphs.

The values of the Steinhardt order parameter shown in FIG. 3A can beused to interpret the results of the MD simulation. A snapshot of atomicpositions and the variation of q _(6,O—O) and q _(6,Al—Al) as a functionof position in MD simulation before crystallization is shown in FIG. 3B.The α-Al₂O₃ phase shown for the calibration of the Steinhardt orderparameter in FIG. 3A had a peak value of q _(6,O—O) of 0.26 and q_(6,Al—Al) of 0.35. The position dependence of q ₆(z) in FIGS. 3A-3C wasobtained by averaging of all atoms within from z to z+4 Å. Thecrystalline region in the simulation (α-Al₂O₃) had values of theseparameters of 0.27 and 0.35, as appropriate for the initial conditionsof the simulation. The values of q _(6,O—O) and q _(6,Al—Al) for theamorphous region of FIG. 3B were also consistent with the initialexpectations.

The phase formed upon crystallization can be determined from theSteinhardt order parameters obtained for the crystallized layers andvisual inspection. FIG. 3C shows a snapshot of the simulation and thespatial variation of q _(6,O—O) and q _(6,Al—Al) after the completion ofthe crystallization simulation. The expected values of q _(6,O—O) and q_(6,Al—Al) for the γ-Al₂O₃ phase were 0.32 and 0.15, respectively. Thecrystallized layer in FIG. 3C had values q _(6,O—O) and q _(6,Al—Al) of0.31 and 0.15, matching γ-Al₂O₃ closely. These values, however, do notmatch the γ-Al₂O₃ or α-Al₂O₃ polymorphs, indicating that the MDsimulation resulted in the epitaxial crystallization of γ-Al₂O₃,consistent with the experiment.

The results shown in FIGS. 3A-3C suggest a mechanism for thecrystallization of amorphous Al₂O₃. The transformation from amorphous toγ-Al₂O₃ can structurally occur by crystallization of the 0 sublatticeinto FCC packing with minimal change in the Al atom arrangement fromfive-fold coordination to six-fold coordination. In this picture,γ-Al₂O₃ forms first out of amorphous Al₂O₃ since the γ-Al₂O₃ polymorphsis structurally more similar to the amorphous structure than γ-Al₂O₃ interms of Al atom arrangement, which can be seen while comparing the q_(6,Al—Al) values in FIG. 3A.

The formation of the γ-Al₂O₃ in the experiments was studied using aseries of x-ray diffraction studies. The crystallographic parameters ofthe γ-Al₂O₃ phase were probed using the γ-Al₂O₃ 202 and 401 families ofx-ray reflections, which arise in a region of reciprocal space in whichthere are no γ-Al₂O₃ reflections. A diagram of the reciprocal space ofthe γ-Al₂O₃ layers formed in these experiments and the x-ray scatteringgeometry are shown in FIG. 4A. The x-ray diffraction patterns acquiredas average during a rotation through the Bragg conditions for thesereflections after heating to 800° C. and 1020° C. are shown in FIGS. 4Band 4C, respectively. The diffraction patterns were obtained by rotatingthe sample over a 5° range of incident angles with respect to the samplesurface. The layer crystallized at 800° C. exhibited only weakreflections of γ-Al₂O₃, which indicates that a small fraction of thefilm was in the γ-Al₂O₃ phase. Two intense pairs of 202 and 401reflections from two variants of γ-Al₂O₃ were apparent in the filmheated at 1020° C., as shown in FIG. 4C.

The angular separation of the reflections in FIG. 4C was measured usingthe diffraction angle χ, which is the azimuthal angle between thereflections along the diffraction cone of constant 20 intercepting theEwald sphere. The intensities integrated across the regions indicated inthe diffraction patterns are shown as a function of χ in FIG. 4D. Theangular separation of the 202 and 401 reflections for the layer heatedto 1020° C. was 45.7°. The interplanar spacings derived from the valuesof 20 and the χ separation in FIG. 4D were a=11.9 Å, c=5.62 Å andβ=103.3°. The lattice parameter b=2.86 Å was derived by combining thesemeasurements with the measured 20 angle of the 712 reflection. Thelattice parameters and χ separation were consistent with the reportedstructure of γ-Al₂O₃. (Guse, W. et al., N. Jahrbuch Mineral. Monat.1990, 217-226.) Twelve-fold azimuthal symmetry was observed in the 712reflections, which also indicated the phase reached by thetransformation from γ-Al₂O₃ was γ-Al₂O₃.

The structural parameters of the γ-Al₂O₃ phase were measured using twosets of x-ray reflections that provided (i) the interplanar spacingalong the surface-normal direction of reciprocal space and (ii) thedistortion in the plane parallel to the surface. FIG. 5A shows θ/2θscans of the films that had been heated to 800° C. and 1020° C. for 1hour. The thin-film reflections in FIG. 5A were indexed with the γ-Al₂O₃and γ-Al₂O₃ reflections along this line in reciprocal space, which hadnearly identical interplanar spacings. As indicated in FIGS. 2A-2C,films heated to 800° C., however, were composed mostly of γ-Al₂O₃ andthe reflections and the intensity peaks at that temperature can beindexed as the γ-Al₂O₃ 111, 222, 333, and, 444 reflections. The (111)interplanar spacing of γ-Al₂O₃ determined from the γ-Al₂O₃ reflectionsin FIG. 5A was 4.53 Å.

An x-ray rocking curve for the γ-Al₂O₃ 222 reflection of a sample heatedto 800° C. is shown in FIG. 5B, along with a rocking curve for a layerof a mixture of the γ-Al₂O₃ and θ-Al₂O₃ phases after heating to 1020° C.The FWHM of the rocking curve of γ-Al₂O₃ was 0.030 after heating at 800°C. It was hypothesized that the larger FWHM of the sample heated at1020° C. arose from the microstructural inhomogeneity associated withthe partial transformation from γ-Al₂O₃ to θ-Al₂O₃.

The interplanar spacing of γ-Al₂O₃ in the plane parallel to the surfacewas measured using the γ-Al₂O₃ 400 family of x-ray reflections. Diagramsof the reciprocal space of γ-Al₂O₃ and θ-Al₂O₃ and the x-ray scatteringgeometry are shown in FIG. 6A. The schematic in FIG. 6A also includesthe α-Al₂O₃ 1123 reflection that serves as a reference for the overallepitaxial relationship. The diffraction patterns in FIGS. 6A-6E wereobtained by integrating the diffracted intensity while rotating thex-ray incident angle through a range spanning the Bragg conditions ofthe γ-Al₂O₃ 400 and θ-Al₂O₃ 600 reflections. The diffraction pattern andintegrated intensity as a function of 2θ are shown in FIGS. 6B and 6Cafter heating to 800° C. and in FIGS. 6D and 6E after heating to 1020°C. The γ-Al₂O₃(400) interplanar spacing determined from the γ-Al₂O₃ 400reflection was 1.98 Å.

The γ-Al₂O₃(111) and (400) interplanar spacings were not preciselyconsistent with cubic symmetry and thus indicate that the γ-Al₂O₃ layerwas elastically distorted. Possible sources of elastic distortioninclude stress remaining after the relaxation of the large epitaxialmismatch between γ-Al₂O₃ and α-Al₂O₃. The hypothetical lattice parameterof undistorted cubic γ-Al₂O₃ can be determined based on the assumptionthat the remaining strain has biaxial symmetry. The interplanar spacingof a purely in-plane γ-Al₂O₃ 422 reflection was determined using thedifference between the γ-Al₂O₃ 222 and 600 reflection vectors. Using thein-plane and out-of-plane interplanar spacings, it was found that thereflections were fit with a cubic γ-Al₂O₃ phase with lattice parameter7.88 Å, in-plane strain 0.55%, and out-of-plane strain −0.39%. Thecalculations employed a Poisson ratio of 0.26 from the literature. (Tu,B. et al., J. Am. Ceram. Soc. 2014, 97, 2996-3003.) Varying the specificchoice of the Poisson ratio over the entire reasonable range of possiblevalues did not have a significant impact on the results.

The orientations and contact planes of the epitaxial films weredetermined from the x-ray diffraction studies. In the θ/2θ scans in FIG.5A, γ-Al₂O₃ 111 and θ-Al₂O₃ 201 reflections appeared along the normaldirection of α-Al₂O₃ (0001) substrate, which indicates that theepitaxial relationship is such that the following planes were parallel:a (0001) 17 (111) 0 (201). The asymmetric reflections of γ-Al₂O₃ 400 andθ-Al₂O₃ 600 appeared in the same azimuthal angle with the same sampletilting as the 1123 reflection of α-Al₂O₃ substrate, as shown in FIG.6D. The following in-plane directions are thus parallel: α[1100]∥γ[011]∥θ[010]. The epitaxial relationship and the bondingconfiguration at the interfaces between the crystallized γ-Al₂O₃ andθ-Al₂O₃ layers and the α-Al₂O₃ substrate are schematically shown inFIGS. 7A and 7B, respectively. Al atoms at tetrahedral and octahedralsites are shaded differently to emphasize the bonding structures and theepitaxial relationships at the interfaces of two phases. The drawingswere constructed based on the assumption that the oxygen sublatticeswere matched at the interfaces.

The azimuthal arrangement of the γ-Al₂O₃ reflections indicates that theγ-Al₂O₃ layer formed with two structural variants. In general, rotationaround the [111] direction of a cubic crystal should yield reflectionswith three-fold azimuthal symmetry. Azimuthal rotation of the filmsproduced by SPE, however, revealed six intensity maxima, which indicatesthat two domains were present in the γ-Al₂O₃ layer. The two domains arerelated by a 1800 rotation around [111], which is consistent with thetwo possible ways that the FCC stacking sequence of the oxygen layerscan be continued from the α-Al₂O₃ substrate into the γ-Al₂O₃ layer.

Sites of Al Vacancies in γ-Al₂O₃ and Variation of the γ-Al₂O₃ LatticeParameter

The conventional unit cell of the spinel structure on which γ-Al₂O₃ isbased includes 32 oxygen-ion sites and 24 sites for cations, consistingof 16 sites with octahedral symmetry and 8 with tetrahedral symmetry.The Al₂O₃ composition requires that each conventional unit cell includesonly 64/3≈21.33 Al cations sties and that the remaining 8/3≈2.67 cationsites be vacant. (Cai, S.-H. et al., Phys. Rev. B 2003, 67, 224104.) Thehypothetical structures in which Al vacancies occupy only octahedralsites or only tetrahedral sites are shown in FIGS. 8A and 8B,respectively.

The lattice parameter of γ-Al₂O₃ varies depending on the sites of Alvacancies. The lattice parameter γ-Al₂O₃ determined from the X-rayanalysis can thus be used to determine the sites of Al vacancies. Theγ-Al₂O₃ lattice parameter ranged from 7.84 Å when Al vacancies were attetrahedral sites to 7.95 Å when Al vacancies were at octahedral sites.(Eberhart, J.-P. Bull. Minéral. 1963, 86, 213-251; Zhou, R.-S. et al.,Acta Cryst. B 1991, 47, 617-630.) A linear interpolation between thesecases indicates that the experimentally observed lattice parameter arosewhen 64% of the Al vacancies sat on sites with tetrahedral coordination.An example of a distribution of vacancies producing a lattice parametermatching the experimentally observed value is shown in FIG. 8C. Five ofthe eight of the Al vacancies in FIG. 8C were at tetrahedral sites,close to the value of 64% matching the experimental lattice parameter.

The structure produced by the MD simulation also provides insight intothe positions of Al vacancies within the partially occupied spinel unitcell. The coordination number distribution of epitaxial γ-Al₂O₃ wascomputed using positions obtained in a 100 ps MD relaxation ofstructures incorporating only octahedral or only tetrahedral vacancysites. When the vacancies were only at octahedral sites, the percentageof Al atoms with octahedral coordination was 62.5%. With vacancies onlyat tetrahedral sites, the same percentage was 75%. The percentage of Alatoms with octahedral coordination for the epitaxial γ-Al₂O₃ produced byMD simulation was 70%, which implies that 78% of the vacancies were attetrahedral sites by comparing to the two extreme percentages of theoccupation sites.

The variation of the lattice parameter of the γ-Al₂O₃ led to theintriguing possibility that the distribution of vacancy sites wasadopted during epitaxy to select a lattice parameter that minimized theelastic energy. The γ-Al₂O₃ on α-Al₂O₃ selected a configuration in thestructural mismatch between epitaxial γ-Al₂O₃ and substrate α-Al₂O₃ thatwas reduced by favoring the vacancies on tetrahedral sites.

The word “illustrative” is used herein to mean serving as an example,instance, or illustration. Any aspect or design described herein as“illustrative” is not necessarily to be construed as preferred oradvantageous over other aspects or designs. Further, for the purposes ofthis disclosure and unless otherwise specified, “a” or “an” can meanonly one or can mean “one or more.” Embodiments consistent with bothconstructions are covered.

The foregoing description of illustrative embodiments of the inventionhas been presented for purposes of illustration and of description. Itis not intended to be exhaustive or to limit the invention to theprecise form disclosed, and modifications and variations are possible inlight of the above teachings or may be acquired from practice of theinvention. The embodiments were chosen and described in order to explainthe principles of the invention and as practical applications of theinvention to enable one skilled in the art to utilize the invention invarious embodiments and with various modifications as suited to theparticular use contemplated. It is intended that the scope of theinvention be defined by the claims appended hereto and theirequivalents.

What is claimed is:
 1. An Al₂O₃ film comprising at least 70 mol. %γ-Al₂O₃, the Al₂O₃ film having an x-ray rocking curve for an Al₂O₃ 222reflection with a full width at half maximum of 0.06° or lower, a filmthickness of less than 200 nm, and a lateral cross-sectional area of atleast 1 cm².
 2. The method of claim 1, wherein the Al₂O₃ film thicknessis 10 nm or less.
 3. The Al₂O₃ film of claim 1, comprising at least 90mol. % γ-Al₂O₃.
 4. The Al₂O₃ film of claim 1, wherein the full width athalf maximum of the x-ray rocking curve for the Al₂O₃ 222 reflection isin the range from 0.010 to 0.060 and the film thickness is in the rangefrom 2 nm to 100 nm.
 5. The Al₂O₃ film of claim 4, comprising at least90 mol. % γ-Al₂O₃.
 6. The Al₂O₃ film of claim 1, wherein the Al₂O₃ filmcomprises only a single γ-Al₂O₃ domain.
 7. The Al₂O₃ film of claim 1,wherein the Al₂O₃ film is disposed on an α-Al₂O₃ substrate.
 8. The Al₂O₃film of claim 7, wherein the Al₂O₃ film is disposed on a (0001) surfaceof the α-Al₂O₃ substrate.
 9. The Al₂O₃ film of claim 7, wherein theα-Al₂O₃ substrate is a miscut substrate and the Al₂O₃ film comprisesonly a single γ-Al₂O₃ domain.
 10. A heterostructure comprising: an Al₂O₃film comprising at least 70 mol. % γ-Al₂O₃, the Al₂O₃ film having anx-ray rocking curve for an Al₂O₃ 222 reflection with a full width athalf maximum of 0.06° or lower, a film thickness of less than 200 nm,and a lateral cross-sectional area of at least 1 cm²; and an epitaxialoverlayer on the Al₂O₃ film, the epitaxial overlayer comprising aninorganic oxide having a cubic crystal structure or a hexagonal crystalstructure.
 11. The heterostructure of claim 10, wherein the Al₂O₃ filmthickness is 10 nm or less.
 12. The heterostructure of claim 10, whereinthe Al₂O₃ film comprises at least 90 mol. % 7-Al₂O₃.
 13. Theheterostructure of claim 10, wherein the full width at half maximum ofthe x-ray rocking curve for the Al₂O₃ 222 reflection is in the rangefrom 0.01° to 0.06° and the film thickness is in the range from 2 nm to100 nm.
 14. The heterostructure of claim 13, wherein the Al₂O₃ filmcomprises at least 90 mol. % 7-Al₂O₃.
 15. The heterostructure of claim10, wherein the Al₂O₃ film comprises only a single 7-Al₂O₃ domain. 16.The heterostructure of claim 10, further comprising an α-Al₂O₃ substratesupporting the Al₂O₃ film.
 17. The heterostructure of claim 16, whereinthe Al₂O₃ film is disposed on a (0001) surface of the α-Al₂O₃ substrate.18. The heterostructure of claim 16, wherein the α-Al₂O₃ substrate is amiscut substrate and the Al₂O₃ film comprises only a single 7-Al₂O₃domain.
 19. The heterostructure of claim 10, wherein the inorganic oxideis Pb[Zr_(x)Ti_(1-x)]O₃, BiFeO₃, KTaO₃, SrVO₃, or a garnet having thegeneral chemical formula X₃Z₂(TO₄)₃, where X, Y, and T are independentlyselected from metal or metalloid elements.